Finite temperature molecular dynamics simulation methods that are driven by the forces obtained “on-the-fly” with an electronic structure theory such as density functional theory are so-called ab-initio molecular dynamics (AIMD). With the pioneering contributions of Roberto Car and Michele Parrinello, the field of AIMD started and became a landmark of computational science. The Car-Parrinello molecular dynamics (CPMD) made AIMD possible to study realistic systems such as dynamics of the solution, phase transition and mass-transportation, etc.

In Born-Oppenheimer molecular dynamics, it solves the energies or forces from the electronic structure theory via minimization or diagonalization methods. At each molecular dynamics step, the electronic structure only relies on the given fixed nuclear positions, namely the electron and nuclei are not coupled. Therefore, the electronic structure problem can be solved by time-independent Schrodinger's equation. And the nuclei are propagated according to classical mechanics or quantum mechanics. The resulting Born-Oppenheimer molecular dynamics method is defined as:

\begin{equation} {\displaystyle M_{I}{\ddot {\mathbf {R} }}_{I}=-\nabla _{I}\,\underset{\{ \psi_0 \}}{\mathrm {min}}\{\langle \psi_0 \mid \mathcal{H}_e \mid \psi_0 \rangle\}} \end{equation} \begin{equation} E_0 \psi_0 = \mathcal{H}_e \psi_0 \end{equation}

The ground state energy can be obtained from any variational method such as density functional theory.

Ab-initio Molecular dynamics simulations are most commonly carried out according to the Born-Oppenheimer procedure, i.e., by optimizing the electronic structure at each step along with the integration of the equations of motion for the atomic coordinates. This protocol can be significantly accelerated by properly propagating the electronic density by ad hoc extrapolation schemes.

In this exercise, we will perform an AIMD simulation for liquid water using the PBE-D3 functional. Regarding the choice of the DFT functional for water, one can refer to this Perspective: How good is DFT for water?